Euler class groups and a theorem of Roitman
The authors define and study the \(n\)th Euler class group \(E^n(A[T])\) of a polynomial ring \(A[T]\) over a commutative regular domain \(A\) such that \(2n\geq\dim A+3\) (M. K. Das, one of the two authors of the present paper, has previously defined \(E^n(A[T])\) for a \(d\)-dimensional commutative Noetherian ring \(A\) containing \(\mathbf Q\) for the case \(d=n\geq 2\)). Among other results, the authors prove that there exists a canonical group homomorphism \(\Phi:E^n(A)\to E^n(A[T])\) that is left invertible. They show that this homomorphism is an isomorphism under some additional assumptions and ask whether this is true in general. The authors prove under certain assumptions, that if the canonical map \(\Phi:E^n(A)\to E^n(A[T])\) is surjective (so \(\Phi\) is an isomorphism), then the map \(\Phi:E^n(A_S)\to E^n(A_S[T])\) is also an isomorphism for every multiplicative subset \(S\) of \(A\); for example, this holds in the above-mentioned Das setting. The authors prove under certain assumptions that if \(I\) is an ideal of \(A[T]\) of height \(n\), \(\underline f=\{f_1,\dots,f_n\}\subseteq I\), and \(I=\underline f+(I^2T)\), then the set \[ N(I;\underline f)=\{s\in A\mid\exists g_1,\dots,g_n\in I_s \text{ such that } I_s= (g_1,\dots,g_n);\;g_i- f_i\in (I^2T)_s\} \] is an ideal, which they call the \textit{Nori ideal} associated to \(I\) and \((f_1,\dots,f_n)\). Several applications are presented, including a ``local global'' principle for the Euler class group, and a new proof of Nori's conjecture on projective generation of ideals; this conjecture was proved by S. M. Bhatwadekar and M. K. Keshari.
scientific article; zbMATH DE number 5882426
scientific article